Download University Physics 226N/231N Old Dominion University Work

University Physics 226N/231N
Old Dominion University
Work, Energy, Power,
and Energy Conservation
Dr. Todd Satogata (ODU/Jefferson Lab)
[email protected]
http://www.toddsatogata.net/2012-ODU
Wednesday, October 3 2012
Happy Birthday to Gwen Stefani, India Arie, Keb’ Mo’, and Stevie Ray Vaughan!
Happy Caramel Custard Day, Techies Day, and Virus Appreciation Day!
Prof. Satogata / Fall 2012
ODU University Physics 226N/231N
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Review: Work
§  Impulse is useful, but another combination is even moreso
§  We define work W as net force applied times the distance its applied in
•  The force can be different at various spots so we really need to add
together (roughly constant) force over small distances
•  This is really another integral, like impulse. It’s a scaler (units: 1 J=1 N-m)
W ≡
�
F dx
Prof. Satogata / Fall 2012
work in one dimension
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Review: Integrals
�
x2
F (x)dx
x1
§  An integral is really a combination of two things:
§  A sum of rectangular areas that are sketched “under” a curve
•  Since these are areas, the integral has the same units as the units
of the x-axis times the units of the y-axis – whatever those are.
•  In physics, we must pay careful attention to units of the axes.
§  A limit of that sum of areas as the width of each rectangle gets
smaller and smaller (and approaches zero)
•  We add together more and more rectangles in this process
•  There are definite and indefinite integrals
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Examples of Work (from a Physicist J )
§  Work is really a bookkeeping tool
§  We’ll relate it to how the energy of an object or system changes
§  We only count force and displacement in the same direction
§  Forces applied perpendicular to Δx or when Δx=0 do no work!
•  Holding an object still against gravity, or moving it horizontally
•  Static frictional forces (since there is no Δx)
Prof. Satogata / Fall 2012
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Work Done Against Gravity
•  The work done by something lifting an object of mass m
against gravity depends only on the vertical distance h:
W = mgh
•  The work is positive if the
object is raised (moved
against the force of gravity)
and negative if it’s lowered
(moved with the force of
gravity).
•  The horizontal motion only
does (positive) work that is
lost to friction
Prof. Satogata / Fall 2012
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Work Done in Stretching a Spring
•  A spring exerts a force Fspring = −kx
•  Someone stretching a spring exerts a force Fstretch = +kx ,
and the work done is
�
� �x
� x
� x
�
1 2
1 2
�
W =
F (x) dx = k
x dx =
kx + c � = kx = W
2
2
0
0
0
Fstretch
•  Here the work is the triangular area under the plot of force
F vs distance stretched x:
distance stretched x
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Energy
§  Energy: the capacity of an object to perform work
§  Energy is what we add up when we do our bookkeeping
§  Work is how energy moves through application of forces
•  We’ll see several examples through this class
•  Think of work as “energy exchange” in the bookkeeping
•  If I “do work”, I lose energy and it goes somewhere else
•  But it doesn’t just disappear
§  How do we do the energy bookkeeping for a
system?
§  Add up energy from a variety of different sources and
things that we know can do work
§  Conservation of energy: total energy for a system
is constant as the system changes over time
•  This appears to be a true fundamental law of nature
•  If energy changes, the bookkeeping wasn’t done right
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Some Energy Bookkeeping Terms
§  Kinetic energy: energy of object’s motion
KE = ½ mv2
velocity v
mass m
§  Gravitational potential energy: energy from the
potential of falling a certain distance under
constant gravity: PEg=mgΔy
m
Δy
m
Δx
§  Spring potential energy: PEs = ½ k(Δx)2
k
§  Energy lost to friction over distance Δx:
Ef = µknΔx
Ffriction = µk n
mass m
Δx
§  Chemical energy, nuclear energy, and others…
Prof. Satogata / Fall 2012
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Work and Net Work
§  Energy: the capacity of an object to perform work
§  Energy is what we add up when we do our bookkeeping
§  Work is how energy moves through application of forces
§  Since work involves transfer of energy, and we want to
account for all energy, it’s important to account for all forces
§  Example: Pulling a box against
friction at constant velocity
§  Net sum of forces on box is zero
ŷ
§  So work done on box is zero
§  But I still do work (I’m exerting a
force over a distance)
§  The energy of my work goes into
frictional losses
Prof. Satogata / Fall 2012
Pulling
force F�pull
Normal force �n
Frictional
force F�f
x̂
m
velocity
�v0
Gravitational
force F�g = mg
Coefficients of friction µs , µk
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Example Friction Problem Revisited
§  Some problems become much easier with conservation of energy
§  Example: A box sliding to a stop due to friction on a surface
§  If it’s got an initial velocity v0, how far does it slide before it stops?
2
§  Initial total energy is all kinetic: Einit = (1/2)mv0
§  It stops when it’s lost all its energy to friction:
Efriction = µk n∆x = µk mg∆x
Apply Conservation of Energy
Normal force �n
Frictional
force F�f
Initial
velocity
m
�v0
ŷ
x̂
Gravitational
force F�g = mg
Coefficients of friction µs , µk
Prof. Satogata / Fall 2012
Einit = Efriction
1
mv02 = µk mg∆x
2
v02
∆x =
2µk g
Still need kinematics to get time to stop..
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Example: Ball Toss
§  Consider your professor tossing
a juggling ball upwards
Total energy: PE+KE=constant
§  I do some work on it to add
energy to the system
§  The system is now the ball!
§  At start, h=0 m and all energy is
kinetic energy
§  As the ball moves up, potential
energy grows and kinetic energy
goes down
§  At top, all energy is potential
energy since v=0 m/s and KE=0 J
§  As the ball comes back down,
potential energy is released and
kinetic energy grows again
Prof. Satogata / Fall 2012
Kinetic energy: KE = ½ mv2
Potential energy: PEg=mgh
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Tangible: Skate Park (20 minutes)
§  Let’s take a look at a fun simulation of a skate park
§  There’s a link on the class web page
§  The next slide has a few questions to answer
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Tangible: Skate Park Questions (20 minutes)
§  Lift the skater and watch their motion back and forth
§  Click the “bar graph” button to see this energy graph è
•  Which energy is biggest when the skater is at peak?
•  Which energy is biggest when the skater is at bottom?
§  Click the “grid” button and note the height of the skater
at peak on both sides
•  Does this change when you move the center blue dot to
change the slopes of both sides of the skater ramp?
§  Click the “Track friction” button and raise the coefficient
of friction a little bit
•  How does the energy chart change? Is energy still
conserved? What if you didn’t count the thermal energy?
§  Play with the plots of energy vs position and vs time
§  Load the “loop.xml” file from the class website (safe it
to the desktop and open with applet File/Open menu
•  This is a loop-the-loop! What height do I drop the skater
from so he just barely completes the loop the loop?
•  What happens to energy if he falls and bounces?
Prof. Satogata / Fall 2012
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Skate Park Loop The Loop
§  We investigated the roller
coaster loop design problem
§  How does energy conservation
help us calculate where to start
the roller coaster so it makes it
safely around the loop?
vsafe =
√
gr
(calculated before)
h
Einit = P Eg = mgh
1
1
2
Etop = mvsafe = mgr
2
2
Einit = Etop
⇒
v
r
h=
2
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Work in Multiple Dimensions
§  Work is adding up (integrating) force along a distance
§  But remember that force perpendicular to the distance moved
does no work
§  So what we’re really doing is adding up the component of force
that is along the direction of motion
F�
W = Fx ∆x = F ∆x cos θ
θ
∆�x
Fx = F cos θ
Force along x
§  This process of taking the “product” of two vectors and
getting a scaler by looking at the component is a dot product
W =
�
F� · d�x =
Prof. Satogata / Fall 2012
�
F dx cos θ
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Power and Energy
•  Power is the rate at which work is done or at which energy is
used or produced. If work ∆W is done in time ∆t, then the
average power over this time is
∆W
P̄ =
(average power)
∆t
•  When the rate changes continuously, the instantaneous
power is
∆W
dW
P = lim
=
∆t→0 ∆t
dt
•  Power is measured in watts (W), with 1 W = 1 J/s.
•  Total work or energy follows from power by multiplying (for
constant power) or integrating (for varying power):
� t2
W = P ∆t or
W =
P dt
t1
Prof. Satogata / Fall 2012
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A Varying Force in Multiple Dimensions
•  In the most general case, an object moves on an arbitrary
path subject to a force whose magnitude and whose
direction relative to the path may vary with position.
•  In that case the integral for the work becomes a line
integral, the limit of the sum of scalar products of
infinitesimally small displacements with the force at each
point.
W =
�
�
r2
�
r1
Prof. Satogata / Fall 2012
F� · �r
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Conservative and Nonconservative Forces
•  A conservative force stores any work done against it, and
can give back the stored work as kinetic energy.
•  For a conservative force, the work done in moving between
two points is independent of the path:
•  Because the work done by a
conservative force is path
independent, the work done in
going around any closed path is
�
zero:
F� · d�r = 0
•  A nonconservative force does not store work done
against it, the work done may depend on path, and the work
done going around a closed path need not be zero.
Prof. Satogata / Fall 2012
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Conservative and Nonconservative Forces
•  Examples of conservative forces include
•  Gravity
•  The static electric force
•  The force of an ideal spring (i.e. no heat generated)
•  Nonconservative forces include
•  Friction
•  The electric force in the presence of changing
magnetism
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Potential Energy
§  The stored work associated with a conservative force is
called potential energy.
§  Potential energy is stored energy that can be released as
kinetic energy.
§  The change in potential energy is defined as the negative of
the work done by a conservative force acting over any path
between two points:
� B
∆UAB = −
F� · d�r
A
§  Potential energy change is independent of path.
§  Only changes in potential energy matter.
§  We’re free to set the zero of potential energy at any convenient
point.
Prof. Satogata / Fall 2012
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Two Common Forms of Potential Energy
•  Gravitational potential energy stores
the work done against gravity:
!U = mg!!y
–  Gravitational potential energy increases
linearly with height y.
–  This reflects the constant gravitational force
near Earth s surface.
•  Elastic potential energy stores the work
done in stretching or compressing springs or
spring-like systems:
U = 12 kx 2
–  Elastic potential energy increases quadratically with
stretch or compression x.
–  This reflects the linearly increasing spring force.
–  Here the zero of potential energy is taken in the
spring’s equilibrium configuration.
Prof. Satogata / Fall 2012
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Conservation of Mechanical Energy
§  By the work-energy theorem, the change in an object s
kinetic energy equals the net work done on the object:
∆K = Wnet
§  When only conservative forces act, the net work is the
negative of the potential-energy change: Wnet = –∆U
§  Therefore when only conservative forces act, any change in
potential energy is compensated by an opposite change in
kinetic energy:
∆K + ∆U = 0
§  Equivalently,
K + U = constant = K0 + U0
§  Both these equations are statements of the law of
conservation of mechanical energy.
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Examples
§  A spring-loaded dart gun
§  What s the dart s speed?
•  K + U = K0 + U0 becomes
1
2
2
mv + 0 = 0 + kx
1
2
2
•  So v = k m x
where x is the initial spring
compression.
Prof. Satogata / Fall 2012
§  A spring and gravity
§  How high does the block go?
•  K + U = K0 + U0 becomes
0 + mgh = 0 + 12 kx 2
•  So
kx 2
h=
2mg
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Potential-Energy Curves
§  Potential-energy curves depict •  Potential-energy curves for a
roller-coaster car with three
a system s potential energy
different total energies:
as a function of position or
other quantities representing
the system s configuration.
§  An object with a given total
energy can be trapped in a
potential well established by
points where its total energy
equals its potential energy.
§  These points are turning
points, beyond which the
object cannot move given its
fixed total energy.
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Potential-Energy Curve for a Molecule
§  Potential-energy curves help determine the structure of
systems, from molecules to engineered systems to stars.
§  The potential-energy curve for a pair of hydrogen atoms
shows potential energy as a function of atomic separation.
–  The minimum in the graph
shows the equilibrium separation
of the H2 molecule.
–  It s convenient to take the zero
of potential energy at infinite
separation.
–  Then negative energies
represent bound states of the
hydrogen molecule.
–  Positive states represent
separated hydrogen atoms.
Prof. Satogata / Fall 2012
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Force and Potential Energy
§  Force is greatest where potential energy increases most
rapidly.
§  Mathematically, the component of force in a given direction
is the negative derivative of the potential energy with
respect to position in that direction:
dU
Fx = !
dx
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Interlude: The Curve of Binding Energy
Peak : Iron (Fe)
Uranium (fission)
Hydrogen (fusion)
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Summary
§  Potential energy is stored energy that can be converted to kinetic
energy.
§  The change in potential energy is the negative of the work done by
a conservative force as an object is moved on any path between
B r
two points:
r
ΔU = − ∫ F ⋅ dr .
A
§  When only conservative forces act, the total mechanical energy K
+ U is conserved:
§  Potential-energy curves describe potential energy as a function
of position or configuration.
§  Force is the negative derivative of potential energy: Fx = –dU/dx.
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Clicker Question 1
§  Suppose it takes the same amount of work to push
a trunk across a rough floor as it does to lift a
weight the same distance straight upward. How do
the amounts of work compare if the trunk and the
weight are moved instead on curved paths
between the same starting and ending points?
The two amounts of work will remain equal to each
other.
B. The amount of work to move the trunk will be greater.
C. The amount of work to move the weight will be greater.
A.
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Clicker Question 2
§  Gravitational force actually decreases with height,
but that decrease is negligible near Earth s
surface. To account for the decrease, how would
the exact value of the potential-energy change
associated with a height change h compare with
mgh, where g is the gravitational acceleration at
Earth s surface?
A.  The potential energy change would be equal to mgh.
B.  The potential energy change would be greater than mgh.
C.  The potential energy change would be less than mgh.
Prof. Satogata / Fall 2012
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Clicker Question 3
§  A bowling ball is tied to the end of a long rope and
suspended from the ceiling. A student stands at
one side of the room, holds the ball to her nose,
and then releases it from rest. Should she duck as
it swings back?
A. She should duck!
B. She does not need to duck.
Prof. Satogata / Fall 2012
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Clicker Question 4
§  This figure shows the potential energy for an
electron in a microelectronic device. From among
the labeled points, find the point where the force
on the electron is greatest.
A. Point A
B. Point B
C. Point C
D. Point D
E. Point E
Prof. Satogata / Fall 2012
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Clicker Question 5
§ 
This figure shows the potential energy for an electron in a
microelectronic device. From among the labeled points, find the
rightmost point where the force on the electron points to the left.
A. Point A
B. Point B
C. Point C
D. Point D
E. Point E
Prof. Satogata / Fall 2012
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Clicker Question 6
§ 
Can a system have negative potential energy?
A. No, because a negative potential energy is
unphysical.
B. No, because the kinetic energy of a system must be
equal to its potential energy.
C. Yes, as long as the total energy remains positive.
D. Yes, as long as the total energy remains negative.
E. Yes, because the choice of the zero of potential
energy is arbitrary.
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Clicker Question 7
§ 
Which of the following statements is true about the total
mechanical energy of a system?
A. It is constant if acted upon by a conservative force.
B. It is constant if acted upon by a nonconservative
force.
C. It is always positive.
D. It is equally divided between its kinetic energy and
potential energy.
Prof. Satogata / Fall 2012
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Clicker Question 8
§  To get to the second floor of the physics building,
student A takes the elevator straight up, while student
B walks up the stairway. Assuming that both students
have the same mass, which of the following
statements is true about their gravitational potential
energy after reaching the second floor?
A.  Student A has greater potential energy because he gets to
the second floor faster.
B.  Student B has greater potential energy because he has
traveled a greater distance.
C.  Both students have the same potential energy.
D.  It is impossible to tell because potential energy depends on
the time elapsed and distance traveled.
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Clicker Question 9
§ 
A spring has a spring constant of 100 N/m. How much potential
energy does it store when stretched by 10 cm?
A. 50 J
B. 10 J
C. 5.0 J
D. 0.50 J
E. 0.10 J
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